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169 Rania Mosaad Saad

Furniture design inspired from .

Dr. Rania Mosaad Saad Assistant Professor, Interior Design and Furniture Department, Faculty of Applied Arts, Helwan University, Cairo, Egypt

Abstract: Keywords: Fractals are a new branch of and art. But what are they really?, How Fractals can they affect on Furniture design?, How to benefit from the formation values and Mandelbrot properties of fractals in Furniture Design?, these were the research problem . Julia Sets This research consists of two axis .The first axe describes the most famous fractals IFS fractals were created, studies the Fractals structure, explains the most important L-system fractals properties and their reflections on furniture design. The second axe applying functional and aesthetic values of deferent Fractals formations in furniture design fractals inspired from them to achieve the research objectives. Furniture Design The research follows the descriptive methodology to describe the fractals kinds and properties, and applied methodology in furniture design inspired from fractals.

Paper received 12th July 2016, Accepted 22th September 2016 , Published 15st of October 2016

nearly identical starting positions, and have real Introduction: world applications in and human creations. Nature is the artist's inspiring since the beginning Some architectures and interior designers turn to of creation. Despite the different artistic trends draw inspiration from the decorative formations, across different eras- ancient and modern- and the geometric and dynamic properties of fractals in artists perception of reality, but that all of these their designs which enriched the field of trends were united in the basic inspiration (the architecture and interior design, and benefited nature). from them by adding functional and aesthetic Mathematicians also had influenced by the values to design. nature's properties. They started from the past to So that the research studies the Fractals structure study the nature and conclusion the mathematics and their reflections to design furniture inspired relationships which explain its properties, and from them. analyses these relationships to get results that benefited the world generally and inspired the Problem: artists specially. 1. How can Fractals affect on Furniture The connection between goes design? back thousands of years. Mathematics has been 2. How to benefit from the formation values used in the design of Gothic cathedrals, Rose and properties of fractals in Furniture windows, oriental rugs, mosaics and tiling's. Design? Geometric forms were fundamental to the cubists Objectives: and many abstract expressionists, and award- 1. Inspiration from the geometric and winning sculptors have used as the basis dynamic properties of fractals in Furniture for their pieces. Dutch artist( M.. Escher Design. )represented , , deformations, 2. Appling functional and aesthetic values of reflections, Platonic solids, spirals, , and deferent Fractals formations in Furniture the hyperbolic plane in his works. http://www.ams.org/mathimagery/ Design. Only in the past century have mathematicians Importance: developed an understanding of such common 1. The research describes and analyzes the phenomena as snowflakes, clouds, coastlines, Fractals types and properties. lightning bolts, rivers, in vegetation, and 2. Explanation the effect of Fractals on the trajectory of molecules in furniture design. (Dalton Allan- 2010). This branch of mathematics 3. Mentioning the applications of Fractals in is known as fractal. Fractals force us to alter our furniture and interior design. view of dimensionality, produce patterns from

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Hypotheses: The fractal is based upon an equation that Designing contemporary furniture inspired from undergoes an or – that is, a fractals adds functional and aesthetic values to repetition of itself. The object does not have to design. exhibit exactly the same structure on all scales, but the same type of structures must be apparent on all Methodology: scales for the structure to be considered a fractal The research follows the descriptive methodology (ERIN PEARSE, 2005). to describe the fractals kinds and properties and Benoit B. Mandelbrot (1924 –2010) was a Polish- applied methodology in furniture design inspired born, French and American mathematician with from fractals. broad interests in practical , especially regarding what he labeled as "the art of roughness" Definition of Fractal: of physical phenomena. He referred to himself as "The term was coined by Benoît Mandelbrot* in a "fractalist"( Mandelbrot, Benoit (2012).He is 1975 and was derived from the Latin fractus recognized for his contribution to the field of meaning "broken" or "fractured. fractal , which included coining the word The fractals are a never-ending . They are "fractal'", as well as developing a theory of created by repeating a simple process over and "roughness and self-similarity" in nature. over in an ongoing loop.( Martin Fractals kinds: (www.FractalFoundation.org- Churchill, 2004) 2009) The fractals are a geometrical shape that can be There are many different kinds of fractal images divided into parts, each of which is a smaller and can be subdivided into several groups. duplicate of the whole (Michael Frame - 2016).

In Nature In Geometry In Algebra

Fig.(1) Aloe Plant Fig.(2) Sierpinsky triangle Fig.(3) 1. Natural Fractals: (Michael Frame- 2016)

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1. 1- Branching: lightning bolts, and river networks. Regardless of Fractals are found all over nature, spanning a huge scale, these patterns are all formed by repeating a range of scales. We find the same patterns again simple branching process. and again, from the tiny branching of our blood A fractal is a picture that tells the story of the vessels and neurons to the branching of trees, process that created it.

Fig.(4) Neurons from the human Fig. (5) our lungs are branching Fig. (6) Lichtenberg " lightning" cortex. The branching of our brain fractals with a surface area~ 100m2 formed by rapidly discharging cells creates the incredibly The similarity to a tree is electrons in Lucite. complex network that is significant, as lungs and trees both responsible for all we perceive, use their large surface areas to imagine, remember. exchange oxygen and co2 .

Fig.(7) river network in China, Fig.(8) Dragon tree, formed by a Fig.(9) fern leaves, formed by formed by erosion from repeated sprout branching, and then each of branching, and then each of the rainfall flowing dowenhill for the pranches branching again, etc. pranches branching again, etc. millions of years Fig.(4:9)Natural branching Fractals (www.FractalFoundation.org-2009) (Karl Richard- 2011) 1-2-Spirals:

Fig.(10) A fossilized ammonite Fig.(11) A hurricane is a self- Fig.(12) A spiral galaxy is the from 300 million years ago. A organizing spiral in the atmosphere, largest natural spiral comprising simple, primitive organism, it built driven by the evaporation and hundreds of billions of stars its spiral shell by adding pieces condensation of sea water. that grow and twist at a constant rate.

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Fig.(13) The plant kingdom is full Fig.(14) The turbulent motion of Fig.(15) Snail shell is a self-similar of spirals. An agave cactus forms fluids creates spirals in systems that forms as a spiral of spirals of its spiral by growing new pieces ranging from a soap film to the spirals. rotated by a fixed angle. Many oceans, atmosphere and the surface other plants form spirals in this of jupiter way, including sunflowers, pinecones, etc. Fig.(10: 15)Natural spirals Fractals Richard- 2011) (www.FractalFoundation.org-2009) (Karl 2. Geometric Fractals:

2-1-IFS ( systems): Fractals geometry, branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, a property called self-similarity or self-symmetry. Purely geometric fractals can be made by repeating a simple process. (www.FractalFoundation.org-2009) Fractals derived from standard geometry by using Fig. (17) Sierpinski triangle iterative transformations on an initial common figure like a straight line (the Cantor dust or the von Koch ), a triangle (the Sierpinski triangle), or a cube (the ).( Erin Pearse-2005) IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy Fig. (18) Menger sponge being transformed by a function (hence "functions system"). (Draves, Scott- 2007).

Fig. (16) von Koch curve Fig. (19) IFS being made with two functions.

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2-2-L-system fractals: ( Stefan Kottwitz-2011) developed in 1968 by Aristid Lindenmayer, a An L-system or Lindenmayer system is consists of Hungarian theoretical biologist and botanist at an alphabet of symbols that can be used to the University of Utrecht. Lindenmayer used L- make strings, a collection of production rules that systems to describe the behavior of plant cells and expand each symbol into some larger string of to model the growth processes of plant symbols, an initial "axiom" string from which to development. L-systems have also been used to begin construction, and a mechanism for model the morphology of a variety of translating the generated strings into geometric organisms and can be used to generate self- structures. L-systems were introduced and similar fractals such as iterated function

Fig. (20) The drawn using an L-system. Fig. (21) Pythagoras tree by L-system. 3. Algebraic Fractals:

"We can also create fractals by repeatedly plane. The equation gives an answer, ‘Znew’ . We calculating a simple equation over and over. plug this back into the equation, as ‘Zold’ and Because the equations must be calculated calculate it again. We are interested in what thousands or millions of times, we need happens for different starting values of ‘C’. to explore them. Algebraic fractals are rarely (www.FractalFoundation.org-2009) drawn or painted by hand. It is usually created Generally, when you square a number, it gets indirectly with the assistance of fractals-generating bigger, and then if you square the answer, it gets software, iterating through three phases: setting bigger still. Eventually, it goes to infinity. This is parameters of appropriate fractals software; the fate of most starting values of ‘C’. However, executing the possibly lengthy calculation; and some values of ‘C’ do not get bigger, but instead evaluating the product." get smaller, or alternate between a set of fixed (www.FractalFoundation.org-2009) values. These are the points inside the Mandelbrot 3-1-The Mandelbrot set: Set, which we color black. Outside the Set, all the We start by plugging a value for the variable ‘C’ values of ‘C’ cause the equation to go to infinity, into the simple equation below. Each complex and the colors are proportional to the speed at number is actually a in a 2-dimensional which they expand. (Karl Richard- 2011)

Fig. (22)Equations of Algebraic fractals Mandelbrot Set

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Mandelbrot set: Self- The same fractal as The same fractal as The same fractal as similarityillustrated by above, magnified 6-fold. above, magnified 100- above, magnified 2000- image enlargements. This Same patterns reappear, fold. fold, where the panel, no magnification. making the exact scale Mandelbrot set fine detail being examined difficult resembles the detail at to determine. low magnification. Fig.(23) zooming in Mandelbrot Set

As mathematical equations, fractals are usually fear and instead become curious. nowhere differentiable.( Edyta Patrzalek -2002), Exploring fractals is fun, and we can play with the An infinite can be conceived of as equations to see what happens. The 4 images winding through space differently from an bellow are algebraic fractals known as Julia Sets. ordinary line, still being a 1-dimensional line yet The first image in the upper left comes from the having a indicating it also same equation as the Mandelbrot Set, Z = Z2 + C. resembles a surface. (Mandelbrot, Benoît B. - When we raise the exponent to Z3 (i.e. Z*Z*Z), 2004). the takes on a 3-fold symmetry, and so 3-2- Julia Sets(Karl Richard- 2011) on. The degree of symmetry always corresponds to When people see the intricate and beautiful the degree of the exponent. patterns produced by equations, they lose their

Fig. (24) Equations of Algebraic fractals (Julia Sets) 3-3-Fractal flame: affine transforms. Fractal flames are a member of the iterated 2. Log-density display instead of linear or function system class of fractals (Mitchell binary (a form of tone mapping) Whitelaw -2004).created by in 1992. 3. Color by structure (i.e. by the recursive path Draves' open-source code was later ported into taken) instead of monochrome or by density. Adobe After Effects and The tone mapping and coloring are designed to translated into the fractal flame editor. display as much of the detail of the fractal as "Fractal flames differ from ordinary iterated possible, which generally results in a more function systems in three ways: aesthetically pleasing image." ( Draves, Scott- 1. Nonlinear functions are iterated in addition to 2007)

Fig.(25) A fractal flame created by the Sheep*. Fig.(26) Fractal flame created in Apophysis * *: a project for (software). generating, downloading, and playing fractal movies . while the screen saver is running

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3-4- Newton fractals, including Nova fractals: Raphson method, which is more normally used as Newton's fractal is a special case of a Julia an approximate method of solving equations.( set. It is a type of fractal derived from the Newton- Aaron Burton-2009)

Newton fractal for Newton fractal for Newton fractal three degree-3 for . for , roots Points in the red coloured by root reached, ( ), basins do not reach a shaded by number of coloured by root root. required reached Fig.(27) shapes of Newton fractal.( Aaron Burton-2009) 3-5- fractals. using spherical coordinates in 2009.( Daniel The is a three-dimensional fractal, White-2009) constructed by Daniel White and Paul Nylander

A ray-traced image of The top of 3D Mandelbulb Zoom in 3D Mandelbulb More Zoom in 3D the 3D Mandelbulb for Mandelbulb the iteration v ↦ v8 + c. Fig.(28) zooming in 3D Mandelbulb 3-6- Julia Fractals :( Paul Bourke - principle as the more traditional Julia set except 2009) that it uses 4 dimensional complex numbers Quaternion Julia fractals are created by the same instead of 2 dimensional complex numbers.

Fig.(29)forms of 4d. Quaternion Julia Sets fractals 4-Properties of fractals a/b=b/(a-b). In the second step a square of the 4-1-Self-Similarity: shorter length ‘b’ is put into the rectangle. Next In geometry though, similar means something very again a square is drawn, this time in the remaining specific. Geometric figures are similar if they part of the ‘a-b’-side length and so on. The have the same shape. In order for one figure to be resulting image is a self-similar spiral of similar to another, you must be able to magnify rectangles, produced by a cascade of self-similar the length of the small figure by the scale factor, proportions. and it will become exactly the same size as the larger figure.(Cynthia Lanius -2007 ) In architecture, interior design and furniture the concept of self-similarity has very often been used in the case of the ‘golden section’. Showing this self-similarity first a rectangle is drawn with the larger side length being ‘a’ and the other ‘b’, with fig.(30) the ‘golden section’. Showing the self- the ‘golden section’ being defined as the relation similarity

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4-2-Iterative Formation: branching in these fractals is a never-ending Below is a picture of a similar iterative process. The four images below are successive operation that is fractal. Take a line segment (see zooms into a detail of the Z = Z2 + C Mandelbrot below) and remove the middle third. What is the Set. Two-fold symmetry branches and becomes 4- resulting figure? That's a more complicated figure. fold, which doubles into 8-fold, and then 16-fold. It's a line segment with a hole in it. The branching process continues forever, and the Repeat the process on that figure. In other number of arms at any level is always a power of words, remove the middle third of both of those 2. sections. This produces an even more complicated figure. Now think of doing this infinitely many times. In fact, this is a famous fractal called Cantor Dust.( Cynthia Lanius -2007) 4-3-Ability of branching: (www.FractalFoundation.org-2009) Just as we find branching fractals in nature, we also find branching within mathematic fractals like Fig.(31) Cantor Dust fractal the Mandelbrot Set. Known as “Bifurcation”,

All of these zooms are just scratching the If we explore other algebraic fractals, we surface of the infinitely complex. Some fractals, find similar patterns and progressions. The two like the Mandelbrot Set, become even more images below are details from the Z = Z3 + C intricate and beautiful the deeper we explore. The Mandelbrot Set. Since the equation involves Z image above exists at a depth of 10176 cubed, the arms now branch in 3-fold symmetry. magnifications! Each of the 3 arms branches into 3 more arms, becoming 9-fold symmetry.This then trifurcates into 27 arms, 81, 243, etc. where the number of arms is always a power of 3. (www.FractalFoundation.org-2009)

Fig. (32) branching within algebraic fractals (Z = Z3 Fig. (33) the Mandelbrot Set at a depth of + C Mandelbrot Set). Known as “Bifurcation” 10176 magnification 4-4- Growing scale: 4-5- Dynamical Systems: .( Tsuneyoshi (www.FractalFoundation.org-2009) Nakayama-2009) Mathematical fractals are infinitely A dynamical system of fractals is the complex. This means we can zoom into them mathematical formalism/rule which describes the forever, and more detail keeps emerging. Or, to deterministic evolution of a point in . look at it another way, as we zoom into the fractals, the original object keeps growing.

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Fig.(34) Fractal Dynamic Field 1. Fractal Dynamic Fields: .( Lori Gardi-2009) and England gathered fractal patterns that were Here is a close up of a Fractal Dynamic Field natural, generated and manmade and starting with a point from the INSIDE of the asked 50 subjects their preferences. The found that Mandelbrot Set. Each point represents one 80 percent of the time, regardless of how the iteration or one time through the feedback loop. If fractals were generated, the subjects liked patterns connected consecutive points with lines so we can that had subtle variations on a recurring theme better understand how this dynamic is being rather than patterns that were monotonously formed.It will noticed that the points from this regular or incomprehensibly random. That work is dynamic appear to be orbiting around and described in Jennifer Ouellette's 2001 story on spiraling into a single point or singularity, within Pollock's Fractals in Discover Magazine. The story the space-time . Also, notice how the quotes an environmental scientist as saying that "lines" connecting the points are getting shorter or our ancestors on the African Savanna may have becoming more compressed as they get closer to become attuned to fractal pattern recognition the singularity at the center of this vortex. So, through their need to know whether the grass was space and time are both contracting toward ruffled by the wind or a stalking lion. This kind of singularity. heritage, some scholars speculate, might help 4-6-Fractals Energy: (Djendji Samardzija-2011) explain why artists, architects, writers and Fractals are artistically shaped energy – vibration- musicians instinctively mimic the fractal patterns imprinted on matter. They are unique artistic found in nature." (Prucia Buscell- 2013) that will bring into your space the energy Taylor, whose scholarship includes the visual spectrum of the universal field with the same of fractals, writes in a Physics World vibration as its own, that is, with which it article "Vision of Beauty" that research with resonates in harmony. This harmony and hundreds of participants over the last decade-his resonance are passed on to the surrounding space and that of others-confirms that people find those and bio-system. mid-range fractals most aesthetically appealing. Creation of fractal energy is a process of weaving Further, he writes, exposure to these pleasing threads between energy and matter. The purpose fractals, the kinds found in clouds, trees, galaxies, of creating energy fractals is to materialize the lungs and neurons, can reduce our physiological energies of our aspirations, wants and needs of the responses to stress and activate brain areas physical, emotional and spiritual levels – energies associated with happiness. and function which make up our Essential Being. are intricately linked. Taylor, now director of the It can become a material basis for the structure of University of Oregon Materials Science Institute, our desires, needs as well as for the growth and describes research in the article on how nano scale development of our inner being. Fractal energy fractal eye implants that mimic and communicate penetrates much deeper into the being than we with neurons might restore the vision of people could possibly imagine. who have lost their sight to macular degeneration. 5- Functional and aesthetic values of Fractals: A university news story reports the implants During (Prucia Buscell) study of Jackson Pollock's would be nano flowers seeded by nano particles of paintings, physicist Richard Taylor began to metal that self-assemble in a natural process. wonder whether the fractal patterns he discovered (Prucia Buscell- 2013) on the paint splattered canvases were the source of Artists have always used geometry in the one or Pollock’s timeless appeal. other way for their works, although they may not "Taylor and perceptual psychologists in Australia have been conscious of that. First the decision has

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Furniture design inspired from fractals 178 to be made if something should be formed in three Samardzija-2011) or two dimensions. Second there are many rules 4. The functioning of energy fractals is based on for composition and proportion found in design principles of vibrational healing, i.e. of and art that are based on dimensions. Thinking of frequency, while its effect is based on one of the most important proportion-rules, resonance. The source of frequency is the namely the ‘golden section’, this turns out to be, as color, light, shape, number, structure, and mentioned before, a fractal . Beside that fractal recurrence, which form the thing that we art can be interpreted as a way for finding the attune ourselves to and harmonize our basics of beauty and harmony that are found in the vibrations with, based on a principle of laws of nature. In this way fractal geometry may harmonic resonance. Human being easily help to explain and prove the ‘rules’ of beauty. attunes to and harmonizes with that vibration "(Wolfgang E. Lorenz – 2003). and comes into resonance with the more "The fractal new geometric art shows powerful or constantly present fields. When surprising kinship to Grand Master paintings or those fields have a beneficial effect, they are Beaux Arts architecture. An obvious reason is that health and supporting. If, however, their classical visual arts, like fractals, involve very effect is of a negative nature, they deplete the many scales of length and favor self- life force and very soon lead to deterioration of similarity"(Wolfgang E. Lorenz – 2003). health and vitality. ( Umberto Mosco-2009) 6- The effect of fractals on furniture design: 5. Exposure to pleasing fractals in design can 1. Fractals have aesthetic formation ( Branka reduce our physiological responses to stress Spehar-2003) (Wolfgang E. Lorenz – and activate brain areas associated with 2003)which add aesthetic values to the design. happiness (functional values) (Prucia Buscell- 2. Fractals have dynamic properties such as 2013). vibrational excitations, transport, and spin 7- Fractals Applications: waves in fractals structures.( Tsuneyoshi The designers had begun copying natural Nakayama-2009) They indicate the movement fractals for inspiration to build successful devices. and energy to the design. Below there are just a few examples of fractals 3. "Every fractal in itself has some energy effect being used in architecture, interior design and just like any color or a picture on the wall has. furniture. However, fractals energy is conceived in such a 7-1-Fractals Applications in Architecture: way as to increase and enhance specific and Fractal geometry may help us to understand given frequencies, depending on the intention and analyze that can be found in towns and purpose of the energetic fractal image. of the Middle Ages but also in cathedrals and They are programmed to emit specific other man-made objects up to these days. It may vibrations – those which you wish to have in also help us to transfer this complexity, which also your office, meeting room or bedroom... to be arises from the development over time, to newly your personal energetic basis and support planned cities and buildings – cities and buildings whether you desire to achieve a physical, may then also be reduced to simpler emotional or spiritual goal." (Djendji (Wolfgang E. Lorenz – 2003)

Fig.(35) An ancient African Fig (36),North Fig (37), Anthony Fig.(38), Cathedrals in settlement ( Ba –lla) of southern Indian temple, Batchelor , South Europe Zambia, Indian Temple. 7-2-Fractals Applications in interior design their formation values and properties of fractals in Interior designers used fractals shapes to design and add functional and aesthetic values to decorate and design interior spaces to benefit from design.

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Fig.(39) cool wall in a Fig.(40) Gramercy Fig.(41)Wall paper – sneakers store: Dear Park Townhouse Fig.(42) Pythagoras Tree fractals Fractal // Yellow Design, Munich Fractals, Design by Fractal Mirrors at wall by Johanna Ek Spain Construction, 7-3-Fractals Applications in furniture: formation values and properties of fractals in Furniture designers design their products design and to add functional and aesthetic values using fractals shapes to benefit from their to design.

Fig.(43) Menger dining table Fig.(44) Homune Table by Michael Fig.(45) Fractals Rotating squares ,Chair by John Brevard Young table by Tom Cecil

Fig(46) The 23 drawers clothing Fig.(47) Mandelbulb 3D Export Fig,(48) the Fractal Table is a table storage by Takeshi Miyakawa fractals table by Nic developed by Platform Wertel Oberfell and Matthias Bär 7-4-Fractals Applications in accessories:

Fig.(49)fractals accessories , Mandelbulb 3d model Source: http://nic022.deviantart.com/art/Mandelbulb-3D

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7-5-Fractals Applications in cinema decoration widely used across the industry to and graphic design create special effects, including fictitious landscapes After Loren Carpenter, co-founder of Pixar and imaginary worlds—such as the Genesis planet Animation Studios, read Benoit sequence in Star Trek II: The Wrath of Khanand the Mandelbrot’s Fractals: Form, Chance, and damaged Death Star in Return of the Jedi. Dimension, he began experimenting with fractals to http://www- make his computer graphics look more realistic. This 03..com/ibm/history/ibm100/us/en/icons/fractal/t technique gave rise to software programs now ransform/

Fig.(50) Checker Stir Fig. (51) Ancient labyrinth Fig.(52) Tipping point by Aurelius Cat , by Theli at/Swiden by Hal Tenny /United States Mark Brady/United States

Fig.(53) At the edge of desert Fig.(55) Asteroidean mechinoderm Fig.(54) Mandelbulb Sphere by gulch by Dainbramage , Brent/ by Graham sym, Graham Symmons fraxialmadness, Jaen/ Francs United States /United Kingdom Suggested projects : spiral fractals: Project 1:Furniture inspired from natural 2. Model 2: branching fractals: Spiral arm sofa inspired from natural snail shell. 1. Model 1: the fractal unit used in design has Self-Similarity, The chair has a branching fractals unit in the back similar iterative operation , growing scale and inspired from branching fractals tree, with self dynamic properties such as vibrational excitations symmetry and growing scale outside. and spin waves in fractals structures. This unit provides us with growing positive energy the design has dynamic energy inspire with through branches growing. growing and containment. The chair made of wood with a hole in back, the fractal unit made of then metal plate cutting by laser and fixed with the chair back from front and back .

Fig,(57) Spiral arm sofa inspired from natural snail shell - designed by the researcher Project 3: Furniture inspired from geometric fractals: 1. Model 3: fractal triangle tables inspired from Sierpinski triangle . Every triangle table top has devided into Fig,(56) the branching fractal chair –designed by small triangles with Self-Similarity ; the group the researcher triangle tables form a hexagonal table with similar Project 2: Furniture inspired from natural iterative operation.

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The design is functional for deferent spaces and purpose . 2. Model 5: Arm chair inspired from fractal geometric shape . The design has self-similarity, similar iterative operation, Growing scale and dynamic energy to up. Upholstery Fabric designed from fractals lines with similar iterative which provide dynamic vibrations.

Fig,(58) fractal triangle tables inspired from Sierpinski triangle .

1. Model 4: Arm chair inspired from (von Koch curve) . The design has self-similarity, similar iterative operation, Ability of branching, Growing scale, Dynamical Systems. Fractals have dynamic properties such as vibrational excitations, transport, and spin waves in fractals structures. They indicate the movement and energy.

Fig,(60) Arm chair inspired from fractal geometric shape 1. Model 6: Panel inspired from fractal geometric shape of squares and circles . it can be used as a sliding door in a bookcase ,or apanel on a wall . The design has Self-Similarity, similar iterative operation, Ability of branching , Growing scale and dynamical systems. Fractals have dynamic properties such as vibrational excitations, transport, and spin waves in fractals structures. They indicate the movement and energy.

Fig,(59) Arm chair inspired from (von Koch curve) .

fig. (61)Panel inspired from fractal geometric shape of squares and circles.

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Model 7: Fractals squares coffee table top . the chair has a geometric fractals unit in the back The fractals unit consists of group of squares inspired from snowflake fractals, with self divided by axis into small symmetric triangles, symmetry and growing scale inside. with deferent hatching inside to provide variety in This unit has centering positive energy and design. attractive dynamic moving through unit growing The design has Self-Similarity, similar iterative inside. and growing scale. The chair made of wood, the fractal unit made of The table has a glass panel on the top to be textile Upholstery Fabric to be comfortable and functional in use. functional use.

Fig. () Fractals squares coffee table top . 3. Model 9: Geometric fractals triangles applying in lighting table, arm chair and a bookcase. The design has central attractive, self-similarity, similar iterative and growing scale. Fractals have dynamic properties such as vibrational excitations and transport in fractals structures. They indicate the movement and energy. Fig. (62) geometric fractals dinning chair 2. Model 8 :

Fig. (63-64)Geometric fractals triangles applying in lighting table and arm chair. made of decorative wood veneer on top . The fractals design has centering positive energy and attractive dynamic moving through unit growing inside.

Fig. (65)Geometric fractals triangles applying in a bookcase. 4. Model 10: Fig. (66) Fractals Simple coffee table has a Simple coffee table has a Geometric fractals shape Geometric fractals.

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Model 11: consists of a rectangle which iterated by 120o 3d.printing geometric fractals chair made of round and then branching into symmetric iteration metal construction and a fractal unit in back and base made of 3d printing polyamide material. The fractal base unit consists of triangle divided into small triangles as shown in fig.() , and had repeated to consists a hexagonal shape . The fractals design has Self-Similarity, similar iterative operation, dynamic properties such as vibrational excitations and transport. They indicate the movement and energy.

. Fig. (69)3d fractals coffee table inspired from 3d mandelbulb 7. Model 14: 3d fractals table inspired from geometric squares. The design has central attractive, self-similarity, similar iterative and growing scale. Fractals have dynamic properties such as vibrational excitations and transport in fractals structures. They indicate the movement and energy. The design made of 3d printing wood or metal material with a glass top.

Fig. (67)3d.printing geometric fractals chair. Project 4: Furniture inspired from 3D fractals:

5. Model 12: 3d fractals coffee table made of group of cuboids with Self-Similarity, similar iterative operation, Growing scale, Dynamical Systems, They indicate vibration and growing energy.

Fig. (70) 3d fractals table inspired from geometric square Results : 1. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. 2. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, Fig. (68)3d fractals coffee table made of group of rivers, coastlines, mountains, clouds, cuboids seashells, hurricanes, spiral galaxies to the 6. Model 13: structure of human lungs, etc. 3d fractals coffee table inspired from 3d 3. Fractals can also created by repeatedly mandelbulb, calculating a simple equation over and over The design made of metal and glass top. It thousands or millions of times by using

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computers to explore them. furniture design. (Natural, geometric and algebraic 4. Algebraic Fractals are rarely drawn or painted elements), and applying functional and aesthetic by hand. It is usually created indirectly with values of deferent Fractals formations in Furniture the assistance of fractals-generating software, Design. So the research had achieved the iterating through three phases: setting objectives.And verified the research hypotheses parameters of appropriate fractals software; that designing contemporary furniture inspired executing the possibly lengthy calculation; from fractals adds functional and aesthetic values and evaluating the product. to design. 5. From the famous properties of fractals: Self- Recommendations Similarity, similar iterative operation, Ability 1. Inspiration from the geometric and dynamic of branching, growing scale, Dynamical properties of fractals in architecture, interior Systems. design and Furniture Design. 6. Fractals have aesthetic formations which add 2. Furniture Designers should benefit from aesthetic values to furniture design. fractals aesthetics to add formation values to 7. Fractals design is one such dynamic process. characterize contemporary furniture. It is a generative device which opens up new 3. Appling functional and aesthetic values of design possibilities. In this context, fractals deferent Fractals formations in Furniture and serve as an effective and useful generative interior Design. design. That adds functional and aesthetic References: values to furniture design, and indicate 1. Aaron Burton-2009 :"Newton’s method and movement and energy. fractals" , 8. Benefits accrue when typology as a design https://www.whitman.edu/Documents/Acade method incorporates dynamic process that mics/Mathematics/burton.pdf) gives rise to infinitely variable generative 2. Branka Spehar & others -2003:" Universal suggestions. aesthetic of fractals", Computers & Graphics 9. Fractals are artistically vibration energy that 27 (2003) 813–820, will bring into our space the energy spectrum http://www2.psy.unsw.edu.au/Users/BNewell of the universal field with the same vibration /SCNT2003.pdf as its own, which it resonates in harmony. 3. Cynthia Lanius -2007 ." Fractals", This harmony and resonance are passed on to http://math.rice.edu/~lanius/fractals/iter.html the surrounding space and bio-system. When 4. Dalton Allan and Brian Dalke -2010," those fields have a beneficial effect, they are Fractals", College of Science, Engineering & health and life supporting. If, however, their Technology , effect is of a negative nature, they deplete the Fractals - Saginaw Valley State University , life force and very soon lead to deterioration https://www.svsu.edu/media/writingcenter/Al of health and vitality. lan-Dalke.pdf 10. The purpose of creating fractals energy is to 5. Daniel White-2009:" The Unraveling of the materialize the energies of our aspirations, real 3D Mandelbulb", wants and needs of the physical, emotional http://www.skytopia.com/project/fractal/man and spiritual levels – energies which make up delbulb.html our Essential Being. 6. (Djendji Samardzija-2011:"the Art of fractals 11. Exposure to pleasing fractals in design can energy", reduce our physiological responses to stress http://www.fraktalneenergije.com/en/energy. and activate brain areas associated with html) happiness. 7. Draves, Scott; Erik Reckase (July 2007). CONCLUSION "The Fractal Flame " (pdf). From the results the research had reached to solve Retrieved 2008-07-17. the problem: had defined the concept of fractals, 8. Edyta Patrzalek -2002 :" Fractals: Useful described their kinds and properties. The research Beauty had explained the effect of Fractals on furniture (General Introduction to Fractal Geometry)", design, mentioned the applications of Fractals in Stan Ackermans Institute, architecture, interior design and furniture. So the IPO, Centre for User-System Interaction, results had achieved the research importance. Eindhoven University of Technology, From the suggested projects, the research had http://www.fractal.org/Bewustzijns- reached the possibility of multiple sources of Besturings-Model/Fractals-Useful- inspiration from fractals and their properties in Beauty.htm

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9. ERIN PEARSE, 2005 :" AN Advisor: O.Univ.Prof. DI Dr.techn. Georg INTRODUCTION TO DIMENSION FRANCK-OBERASPACH (PDF Download THEORY AND FRACTAL GEOMETRY: Available).: FRACTAL DIMENSIONS AND https://www.researchgate.net/publication/273 MEASURES" , 457875_Fractals_and_Fractal_Architecture http://www.math.cornell.edu/~erin/docs/dimension.pdf [accessed May 20, 2016]. 10. Gabriele A. Losa and others-2016: " From 19. Paul Bourke -2009:" 4D Quaternion Julia Set Fractal Geometry to ", Ray Tracer", Applied Mathematics, 2016, 7, 346-354 http://2008.sub.blue/blog/2009/9/20/quaternio ,Published Online March 2016 in SciRes. n_julia.html http://www.scirp.org/journal/am 20. Prucia Buscell- 2013: 11. Karl Richard- 2011 :" The Universe ‘May’ http://www.plexusinstitute.org/blogpost/6567 Have A Fractal Structure!" , 63/160028/Fractals-Function-and-Beauty https://polynomial.me.uk/tag/fractals/ 21. Stefan Kottwitz-2011:" Example: 12. Lori Gardi -2009 :" Event Horizons, black Lindenmayer systems", holes and the Mandelbrot set – close to the http://www.texample.net/tikz/examples/linde edge ", nmayer-systems/ http://www.butterflyeffect.ca/Close/Pages/Fra 22. Tsuneyoshi Nakayama-2009:" Fractal ctalDynamicFields.html Structures in Condensed Matter Physics", 13. Mandelbrot, Benoît B. -2004:" Fractals and Toyota Physical and Chemical Research Chaos". Berlin: Springer. p. 38. Institute, Nagakute, Japan. http://subutu- 14. Mandelbrot, Benoit (2012). The Fractalist: ap.eng.hokudai.ac.jp/nakayama/pdf/Encyclop Memoir of a Scientific Maverick, Pantheon edia_of_Complexity_and_Systems_Science_ Books. ISBN 978-0-307-38991-6 Springer-Verlag(2009).pdf 15. Martin Churchill, 2004: "Introduction to 23. Umberto Mosco-2009 :" The Relationship Fractal Geometry", Between Fractal Geometry and Energy" http://mdc.nfshost.com/fractals.pdf ,https://www.wpi.edu/academics/math/fractal 16. Michael Frame and others- 2016 :" Fractal -geometry.html) Geometry-fractals, Graphics, &Mathematics 26. http://www.ams.org/mathimagery/ Education ) , Yale University, 27. http://nic022.deviantart.com/art/Mandelbulb- http://users.math.yale.edu/public_html/People 3D /frame/Fractals/ 28. http://www- 17. Mitchell Whitelaw (2004). Metacreation: Art 03.ibm.com/ibm/history/ibm100/us/en/icons/f and Artificial Life. MIT Press. pp 155. ractal/transform/ 18. Wolfgang E. Lorenz – 2003:" Fractals and 29. www.FractalFoundation.org-2009 "Fractal Fractal Architecture" , Diplom-Ingenieur, Pack 1 Educators’ Guide".

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